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This article explores the less examined aspect of Riemann's Skizze, the Schematirisung der Dissonanzen. The article focuses in particular on Riemann's category of Doppelklänge, by examining the conceptual origin of the category in the Skizze and its relation to Riemann's harmonic theories. Doppelklänge are dissonant chords that arise through a combination of two consonant triads in a coordinate relationship. While the Doppelklänge category is short-lived, it nevertheless fascinating for its relation to Riemann's taxonomy of harmonic relations. Specifically, Riemann classifies Doppelklänge acc...

This article explores the less examined aspect of Riemann's Skizze, the Schematirisung der Dissonanzen. The article focuses in particular on Riemann's category of Doppelklänge, by examining the conceptual origin of the category in the Skizze and its relation to Riemann's harmonic theories. Doppelklänge are dissonant chords that arise through a combination of two consonant triads in a coordinate relationship. While the Doppelklänge category is short-lived, it nevertheless fascinating for its relation to Riemann's taxonomy of harmonic relations. Specifically, Riemann classifies Doppelklänge according to the Harmonieschritte that relate their component Klänge, reflecting within those dissonant chordal structures the same relations that organize harmonic progression in music. The article then examines how an explicitly transformational reinterpretation of Riemann's Doppelklänge can offer analytic insights into tonal, post-tonal and transitional musical repertoires such as by Chopin, William Grant Still, and Ravel. The article ends by providing an appendix that presents a translation of Schematirisung der Dissonanzen.